Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [120,2,Mod(59,120)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("120.59"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(120, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 120 = 2^{3} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 120.m (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.958204824255\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 24x^{14} + 192x^{12} + 672x^{10} + 1092x^{8} + 880x^{6} + 352x^{4} + 64x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{13} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 59.3
Root \(-1.05636i\) of defining polynomial
Character \(\chi\) \(=\) 120.59
Dual form 120.2.m.b.59.2

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.30656 + 0.541196i) q^{2} +(0.541196 - 1.64533i) q^{3} +(1.41421 - 1.41421i) q^{4} +(-1.25928 - 1.84776i) q^{5} +(0.183339 + 2.44262i) q^{6} -3.29066 q^{7} +(-1.08239 + 2.61313i) q^{8} +(-2.41421 - 1.78089i) q^{9} +(2.64533 + 1.73270i) q^{10} -2.51856i q^{11} +(-1.56148 - 3.09221i) q^{12} +4.65369 q^{13} +(4.29945 - 1.78089i) q^{14} +(-3.72169 + 1.07193i) q^{15} -4.00000i q^{16} +3.69552 q^{17} +(4.11813 + 1.02028i) q^{18} +0.828427 q^{19} +(-4.39402 - 0.832235i) q^{20} +(-1.78089 + 5.41421i) q^{21} +(1.36303 + 3.29066i) q^{22} -2.61313i q^{23} +(3.71366 + 3.19510i) q^{24} +(-1.82843 + 4.65369i) q^{25} +(-6.08034 + 2.51856i) q^{26} +(-4.23671 + 3.00836i) q^{27} +(-4.65369 + 4.65369i) q^{28} +6.08034 q^{29} +(4.28250 - 3.41471i) q^{30} -1.17157i q^{31} +(2.16478 + 5.22625i) q^{32} +(-4.14386 - 1.36303i) q^{33} +(-4.82843 + 2.00000i) q^{34} +(4.14386 + 6.08034i) q^{35} +(-5.93277 + 0.895653i) q^{36} +1.92762 q^{37} +(-1.08239 + 0.448342i) q^{38} +(2.51856 - 7.65685i) q^{39} +(6.19146 - 1.29066i) q^{40} +8.59890i q^{41} +(-0.603305 - 8.03782i) q^{42} -6.01673i q^{43} +(-3.56178 - 3.56178i) q^{44} +(-0.250486 + 6.70353i) q^{45} +(1.41421 + 3.41421i) q^{46} +2.61313i q^{47} +(-6.58132 - 2.16478i) q^{48} +3.82843 q^{49} +(-0.129605 - 7.06988i) q^{50} +(2.00000 - 6.08034i) q^{51} +(6.58132 - 6.58132i) q^{52} -4.59220i q^{53} +(3.90742 - 6.22351i) q^{54} +(-4.65369 + 3.17157i) q^{55} +(3.56178 - 8.59890i) q^{56} +(0.448342 - 1.36303i) q^{57} +(-7.94435 + 3.29066i) q^{58} +2.51856i q^{59} +(-3.74732 + 6.77920i) q^{60} -8.48528i q^{61} +(0.634051 + 1.53073i) q^{62} +(7.94435 + 5.86030i) q^{63} +(-5.65685 - 5.65685i) q^{64} +(-5.86030 - 8.59890i) q^{65} +(6.15188 - 0.461750i) q^{66} +3.29066i q^{67} +(5.22625 - 5.22625i) q^{68} +(-4.29945 - 1.41421i) q^{69} +(-8.70487 - 5.70171i) q^{70} +7.12356 q^{71} +(7.26682 - 4.38102i) q^{72} -6.58132i q^{73} +(-2.51856 + 1.04322i) q^{74} +(6.66732 + 5.52692i) q^{75} +(1.17157 - 1.17157i) q^{76} +8.28772i q^{77} +(0.853202 + 11.3672i) q^{78} +16.4853i q^{79} +(-7.39104 + 5.03712i) q^{80} +(2.65685 + 8.59890i) q^{81} +(-4.65369 - 11.2350i) q^{82} -9.37011 q^{83} +(5.13829 + 10.1754i) q^{84} +(-4.65369 - 6.82843i) q^{85} +(3.25623 + 7.86123i) q^{86} +(3.29066 - 10.0042i) q^{87} +(6.58132 + 2.72607i) q^{88} -5.03712i q^{89} +(-3.30065 - 8.89414i) q^{90} -15.3137 q^{91} +(-3.69552 - 3.69552i) q^{92} +(-1.92762 - 0.634051i) q^{93} +(-1.41421 - 3.41421i) q^{94} +(-1.04322 - 1.53073i) q^{95} +(9.77048 - 0.733355i) q^{96} +2.72607i q^{97} +(-5.00208 + 2.07193i) q^{98} +(-4.48528 + 6.08034i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{9} + 16 q^{10} - 32 q^{19} - 32 q^{24} + 16 q^{25} + 16 q^{30} - 32 q^{34} - 32 q^{36} + 32 q^{40} + 16 q^{49} + 32 q^{51} + 32 q^{54} + 64 q^{66} - 64 q^{70} + 32 q^{75} + 64 q^{76} - 48 q^{81}+ \cdots + 64 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/120\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(41\) \(61\) \(97\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.30656 + 0.541196i −0.923880 + 0.382683i
\(3\) 0.541196 − 1.64533i 0.312460 − 0.949931i
\(4\) 1.41421 − 1.41421i 0.707107 − 0.707107i
\(5\) −1.25928 − 1.84776i −0.563167 − 0.826343i
\(6\) 0.183339 + 2.44262i 0.0748477 + 0.997195i
\(7\) −3.29066 −1.24375 −0.621876 − 0.783116i \(-0.713629\pi\)
−0.621876 + 0.783116i \(0.713629\pi\)
\(8\) −1.08239 + 2.61313i −0.382683 + 0.923880i
\(9\) −2.41421 − 1.78089i −0.804738 − 0.593630i
\(10\) 2.64533 + 1.73270i 0.836526 + 0.547927i
\(11\) − 2.51856i − 0.759374i −0.925115 − 0.379687i \(-0.876032\pi\)
0.925115 − 0.379687i \(-0.123968\pi\)
\(12\) −1.56148 − 3.09221i −0.450760 − 0.892645i
\(13\) 4.65369 1.29070 0.645351 − 0.763886i \(-0.276711\pi\)
0.645351 + 0.763886i \(0.276711\pi\)
\(14\) 4.29945 − 1.78089i 1.14908 − 0.475963i
\(15\) −3.72169 + 1.07193i −0.960936 + 0.276771i
\(16\) − 4.00000i − 1.00000i
\(17\) 3.69552 0.896295 0.448147 − 0.893960i \(-0.352084\pi\)
0.448147 + 0.893960i \(0.352084\pi\)
\(18\) 4.11813 + 1.02028i 0.970653 + 0.240483i
\(19\) 0.828427 0.190054 0.0950271 − 0.995475i \(-0.469706\pi\)
0.0950271 + 0.995475i \(0.469706\pi\)
\(20\) −4.39402 − 0.832235i −0.982532 − 0.186093i
\(21\) −1.78089 + 5.41421i −0.388622 + 1.18148i
\(22\) 1.36303 + 3.29066i 0.290600 + 0.701571i
\(23\) − 2.61313i − 0.544874i −0.962174 − 0.272437i \(-0.912170\pi\)
0.962174 − 0.272437i \(-0.0878297\pi\)
\(24\) 3.71366 + 3.19510i 0.758049 + 0.652198i
\(25\) −1.82843 + 4.65369i −0.365685 + 0.930739i
\(26\) −6.08034 + 2.51856i −1.19245 + 0.493930i
\(27\) −4.23671 + 3.00836i −0.815356 + 0.578960i
\(28\) −4.65369 + 4.65369i −0.879465 + 0.879465i
\(29\) 6.08034 1.12909 0.564546 − 0.825402i \(-0.309051\pi\)
0.564546 + 0.825402i \(0.309051\pi\)
\(30\) 4.28250 − 3.41471i 0.781873 − 0.623437i
\(31\) − 1.17157i − 0.210421i −0.994450 − 0.105210i \(-0.966448\pi\)
0.994450 − 0.105210i \(-0.0335516\pi\)
\(32\) 2.16478 + 5.22625i 0.382683 + 0.923880i
\(33\) −4.14386 − 1.36303i −0.721353 − 0.237274i
\(34\) −4.82843 + 2.00000i −0.828068 + 0.342997i
\(35\) 4.14386 + 6.08034i 0.700440 + 1.02777i
\(36\) −5.93277 + 0.895653i −0.988796 + 0.149276i
\(37\) 1.92762 0.316899 0.158450 − 0.987367i \(-0.449350\pi\)
0.158450 + 0.987367i \(0.449350\pi\)
\(38\) −1.08239 + 0.448342i −0.175587 + 0.0727306i
\(39\) 2.51856 − 7.65685i 0.403292 − 1.22608i
\(40\) 6.19146 − 1.29066i 0.978956 − 0.204071i
\(41\) 8.59890i 1.34292i 0.741039 + 0.671461i \(0.234333\pi\)
−0.741039 + 0.671461i \(0.765667\pi\)
\(42\) −0.603305 − 8.03782i −0.0930920 − 1.24026i
\(43\) − 6.01673i − 0.917542i −0.888554 − 0.458771i \(-0.848290\pi\)
0.888554 − 0.458771i \(-0.151710\pi\)
\(44\) −3.56178 − 3.56178i −0.536959 − 0.536959i
\(45\) −0.250486 + 6.70353i −0.0373403 + 0.999303i
\(46\) 1.41421 + 3.41421i 0.208514 + 0.503398i
\(47\) 2.61313i 0.381164i 0.981671 + 0.190582i \(0.0610374\pi\)
−0.981671 + 0.190582i \(0.938963\pi\)
\(48\) −6.58132 − 2.16478i −0.949931 − 0.312460i
\(49\) 3.82843 0.546918
\(50\) −0.129605 − 7.06988i −0.0183289 − 0.999832i
\(51\) 2.00000 − 6.08034i 0.280056 − 0.851418i
\(52\) 6.58132 − 6.58132i 0.912664 − 0.912664i
\(53\) − 4.59220i − 0.630787i −0.948961 − 0.315394i \(-0.897863\pi\)
0.948961 − 0.315394i \(-0.102137\pi\)
\(54\) 3.90742 − 6.22351i 0.531732 − 0.846912i
\(55\) −4.65369 + 3.17157i −0.627504 + 0.427655i
\(56\) 3.56178 − 8.59890i 0.475963 − 1.14908i
\(57\) 0.448342 − 1.36303i 0.0593843 − 0.180538i
\(58\) −7.94435 + 3.29066i −1.04314 + 0.432085i
\(59\) 2.51856i 0.327889i 0.986470 + 0.163944i \(0.0524217\pi\)
−0.986470 + 0.163944i \(0.947578\pi\)
\(60\) −3.74732 + 6.77920i −0.483778 + 0.875191i
\(61\) − 8.48528i − 1.08643i −0.839594 − 0.543214i \(-0.817207\pi\)
0.839594 − 0.543214i \(-0.182793\pi\)
\(62\) 0.634051 + 1.53073i 0.0805245 + 0.194403i
\(63\) 7.94435 + 5.86030i 1.00089 + 0.738329i
\(64\) −5.65685 − 5.65685i −0.707107 − 0.707107i
\(65\) −5.86030 − 8.59890i −0.726881 − 1.06656i
\(66\) 6.15188 − 0.461750i 0.757244 − 0.0568375i
\(67\) 3.29066i 0.402018i 0.979589 + 0.201009i \(0.0644220\pi\)
−0.979589 + 0.201009i \(0.935578\pi\)
\(68\) 5.22625 − 5.22625i 0.633776 − 0.633776i
\(69\) −4.29945 − 1.41421i −0.517593 − 0.170251i
\(70\) −8.70487 − 5.70171i −1.04043 − 0.681485i
\(71\) 7.12356 0.845412 0.422706 − 0.906267i \(-0.361080\pi\)
0.422706 + 0.906267i \(0.361080\pi\)
\(72\) 7.26682 − 4.38102i 0.856403 − 0.516308i
\(73\) − 6.58132i − 0.770285i −0.922857 − 0.385142i \(-0.874152\pi\)
0.922857 − 0.385142i \(-0.125848\pi\)
\(74\) −2.51856 + 1.04322i −0.292777 + 0.121272i
\(75\) 6.66732 + 5.52692i 0.769875 + 0.638194i
\(76\) 1.17157 − 1.17157i 0.134389 − 0.134389i
\(77\) 8.28772i 0.944473i
\(78\) 0.853202 + 11.3672i 0.0966061 + 1.28708i
\(79\) 16.4853i 1.85474i 0.374147 + 0.927370i \(0.377936\pi\)
−0.374147 + 0.927370i \(0.622064\pi\)
\(80\) −7.39104 + 5.03712i −0.826343 + 0.563167i
\(81\) 2.65685 + 8.59890i 0.295206 + 0.955434i
\(82\) −4.65369 − 11.2350i −0.513914 − 1.24070i
\(83\) −9.37011 −1.02850 −0.514252 − 0.857639i \(-0.671930\pi\)
−0.514252 + 0.857639i \(0.671930\pi\)
\(84\) 5.13829 + 10.1754i 0.560634 + 1.11023i
\(85\) −4.65369 − 6.82843i −0.504764 − 0.740647i
\(86\) 3.25623 + 7.86123i 0.351128 + 0.847699i
\(87\) 3.29066 − 10.0042i 0.352796 − 1.07256i
\(88\) 6.58132 + 2.72607i 0.701571 + 0.290600i
\(89\) − 5.03712i − 0.533934i −0.963706 − 0.266967i \(-0.913979\pi\)
0.963706 − 0.266967i \(-0.0860214\pi\)
\(90\) −3.30065 − 8.89414i −0.347919 − 0.937525i
\(91\) −15.3137 −1.60531
\(92\) −3.69552 − 3.69552i −0.385284 − 0.385284i
\(93\) −1.92762 − 0.634051i −0.199885 − 0.0657480i
\(94\) −1.41421 − 3.41421i −0.145865 − 0.352149i
\(95\) −1.04322 − 1.53073i −0.107032 − 0.157050i
\(96\) 9.77048 − 0.733355i 0.997195 − 0.0748477i
\(97\) 2.72607i 0.276790i 0.990377 + 0.138395i \(0.0441944\pi\)
−0.990377 + 0.138395i \(0.955806\pi\)
\(98\) −5.00208 + 2.07193i −0.505286 + 0.209297i
\(99\) −4.48528 + 6.08034i −0.450788 + 0.611097i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 120.2.m.b.59.3 yes 16
3.2 odd 2 inner 120.2.m.b.59.13 yes 16
4.3 odd 2 480.2.m.b.239.7 16
5.2 odd 4 600.2.b.i.251.5 16
5.3 odd 4 600.2.b.i.251.12 16
5.4 even 2 inner 120.2.m.b.59.14 yes 16
8.3 odd 2 inner 120.2.m.b.59.1 ✓ 16
8.5 even 2 480.2.m.b.239.8 16
12.11 even 2 480.2.m.b.239.12 16
15.2 even 4 600.2.b.i.251.11 16
15.8 even 4 600.2.b.i.251.6 16
15.14 odd 2 inner 120.2.m.b.59.4 yes 16
20.3 even 4 2400.2.b.i.2351.1 16
20.7 even 4 2400.2.b.i.2351.16 16
20.19 odd 2 480.2.m.b.239.9 16
24.5 odd 2 480.2.m.b.239.11 16
24.11 even 2 inner 120.2.m.b.59.15 yes 16
40.3 even 4 600.2.b.i.251.8 16
40.13 odd 4 2400.2.b.i.2351.2 16
40.19 odd 2 inner 120.2.m.b.59.16 yes 16
40.27 even 4 600.2.b.i.251.9 16
40.29 even 2 480.2.m.b.239.10 16
40.37 odd 4 2400.2.b.i.2351.15 16
60.23 odd 4 2400.2.b.i.2351.3 16
60.47 odd 4 2400.2.b.i.2351.14 16
60.59 even 2 480.2.m.b.239.6 16
120.29 odd 2 480.2.m.b.239.5 16
120.53 even 4 2400.2.b.i.2351.4 16
120.59 even 2 inner 120.2.m.b.59.2 yes 16
120.77 even 4 2400.2.b.i.2351.13 16
120.83 odd 4 600.2.b.i.251.10 16
120.107 odd 4 600.2.b.i.251.7 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.2.m.b.59.1 ✓ 16 8.3 odd 2 inner
120.2.m.b.59.2 yes 16 120.59 even 2 inner
120.2.m.b.59.3 yes 16 1.1 even 1 trivial
120.2.m.b.59.4 yes 16 15.14 odd 2 inner
120.2.m.b.59.13 yes 16 3.2 odd 2 inner
120.2.m.b.59.14 yes 16 5.4 even 2 inner
120.2.m.b.59.15 yes 16 24.11 even 2 inner
120.2.m.b.59.16 yes 16 40.19 odd 2 inner
480.2.m.b.239.5 16 120.29 odd 2
480.2.m.b.239.6 16 60.59 even 2
480.2.m.b.239.7 16 4.3 odd 2
480.2.m.b.239.8 16 8.5 even 2
480.2.m.b.239.9 16 20.19 odd 2
480.2.m.b.239.10 16 40.29 even 2
480.2.m.b.239.11 16 24.5 odd 2
480.2.m.b.239.12 16 12.11 even 2
600.2.b.i.251.5 16 5.2 odd 4
600.2.b.i.251.6 16 15.8 even 4
600.2.b.i.251.7 16 120.107 odd 4
600.2.b.i.251.8 16 40.3 even 4
600.2.b.i.251.9 16 40.27 even 4
600.2.b.i.251.10 16 120.83 odd 4
600.2.b.i.251.11 16 15.2 even 4
600.2.b.i.251.12 16 5.3 odd 4
2400.2.b.i.2351.1 16 20.3 even 4
2400.2.b.i.2351.2 16 40.13 odd 4
2400.2.b.i.2351.3 16 60.23 odd 4
2400.2.b.i.2351.4 16 120.53 even 4
2400.2.b.i.2351.13 16 120.77 even 4
2400.2.b.i.2351.14 16 60.47 odd 4
2400.2.b.i.2351.15 16 40.37 odd 4
2400.2.b.i.2351.16 16 20.7 even 4